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In this paper, we obtain Stein operator for sum of n independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature.Finally, an application of our results to a function of waiting time for (k1, k2)-events is given.

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In this paper, we consider the sums of non-negative integer valued m-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein operator, uniform and non-uniform bounds on the solution of Stein equation, and etc. Using Stein's method, we obtain the error bounds for the approximation problem considered. As special cases, we discuss two applications, namely, 2-runs and (k 1 , k 2 )runs and compare the bound with the existing bounds.